#6369795

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,4)(5,6,8,7)(9,10)(11,15)(12,16)(13,18)(14,17)(19,25)(20,27)(21,26)(22,28)(23,29)(24,30)(31,35)(32,36)(33,37)(34,38)(39,40);(2,4)(9,11,13,20,14,21,24,33)(10,15,18,27,17,26,30,37)(12,19,23,32,22,31,34,39)(16,25,29,36,28,35,38,40);(5,7,8,6)(9,12)(10,16)(11,19)(13,23)(14,22)(15,25)(17,28)(18,29)(20,32)(21,31)(24,34)(26,35)(27,36)(30,38)(33,39)(37,40);(12,22)(16,28)(19,31)(23,34)(25,35)(29,38)(32,39)(36,40);(10,17)(15,26)(16,28)(18,30)(25,35)(27,37)(29,38)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,h^2=i,b^a=bf,e^a=ei,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276991);